Nonlinear Wave Motion

非线性波动

• 批准号: 2306290
• 负责人: Mark Ablowitz
• 金额: $ 20万
• 依托单位: University of Colorado at Boulder
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-09-01 至 2026-08-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2306290&HistoricalAwards=false
• 关键词: Nonlinear; Wave; Motion

Wave motion is fundamental in nature: Light waves impact the way people see, sound waves allow people to hear, and water waves are critical to the motion of ships and tidal energy. This project is focused on nonlinear wave propagation, that is the study of large amplitude or high-power signals, and aims to enhance the ability of scientists to develop insight and understanding of large amplitude waves and of the associated dynamics of wave energy transfer. The results of the investigation will benefit applications of nonlinear wave motion, which include tsunamis, rogue or freak waves, high power lasers, and the propagation of large changes in pressure waves such as those that occur in shock waves and blast waves.For linear waves or small amplitude waves a substantial theory is available and, in many situations, exact solutions can be found. For nonlinear waves the available theory is less developed and there are relatively few nonlinear wave equations for which exact solutions are known. However, the so-called Inverse Scattering Transform method has been used to find solutions and understand important properties of the underlying equations and has led to a wide class of solutions of physically significant nonlinear wave equations. This method can be used to find so-called solitons solutions, stable localized waves which have special interaction properties and have been found to apply widely, for example in water waves, electromagnetic waves, lattice dynamics, magnetic waves, and elasticity. The investigator will consider extensions of this method to new classes of nonlinear equations, such as fractional nonlinear wave equations and their soliton solutions. Fractional equations themselves have numerous applications, including to the study of amorphous materials. Additionally, the method will be applied to study the properties of a class of multidimensional solitons, that is lump-type waves that vanish in all directions, which are relevant also for equations that arise in quantum mechanics and optics.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

波浪运动是自然界的基础：光波影响人们的视觉方式，声波让人们听到声音，而水波对于船舶和潮汐能的运动至关重要。 该项目专注于非线性波传播，即大振幅或高功率信号的研究，旨在增强科学家对大振幅波以及波能量传递相关动力学的洞察和理解的能力。研究结果将有利于非线性波动的应用，包括海啸、异常波、高功率激光，以及压力波大变化的传播，例如冲击波和爆炸波中发生的波动。对于线性波或小振幅波，可以使用实质性理论，并且在许多情况下可以找到精确的解决方案。对于非线性波，可用的理论还不太发达，并且已知精确解的非线性波动方程相对较少。然而，所谓的逆散射变换方法已被​​用来寻找解并理解基础方程的重要性质，并导致了一系列具有物理意义的非线性波动方程的解。这种方法可用于寻找所谓的孤子解，即稳定的局域波，它们具有特殊的相互作用特性，并已被发现广泛应用，例如在水波、电磁波、晶格动力学、磁波和弹性中。研究人员将考虑将该方法扩展到新类别的非线性方程，例如分数非线性波动方程及其孤子解。分数方程本身有许多应用，包括非晶材料的研究。此外，该方法将用于研究一类多维孤子的性质，即在所有方向上消失的块型波，这也与量子力学和光学中出现的方程相关。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。